$$U_{1243}(x, y) = \frac{\left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{y^{4} \left(- \frac{x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)^{2}}{y^{4}} + 4\right)}$$

$$T_{1243}(n, m, k) = \frac{\left(2 k + 2 n\right) {\binom{k + n - 1}{n}} {\binom{4 k + 2 m + 4 n}{m}}}{2 k + m + 2 n}$$